Tate's conjecture

Tate's conjecture is a mathematical idea suggesting a deep connection between certain geometric shapes called algebraic varieties and their associated symmetries, captured by Galois groups. Specifically, it predicts that the important algebraic features of these shapes—like their "cycles" or patterns—can be fully described by the way their points behave under these symmetries. In essence, it links the geometry of shapes with number theory, proposing that understanding symmetries can reveal the intrinsic geometric properties of these varieties. This conjecture remains one of the central open problems in modern algebraic geometry and number theory.